Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-x-6y &= -1 \\ -2x-9y &= -6\end{align*}$
Explanation: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-2$ and the bottom equation by $1$ $\begin{align*}2x+12y &= 2\\ -2x-9y &= -6\end{align*}$ Add the top and bottom equations. $3y = -4$ Divide both sides by $3$ and reduce as necessary. $y = -\dfrac{4}{3}$ Substitute $-\dfrac{4}{3}$ for $y$ in the top equation. $-x-6( -\dfrac{4}{3}) = -1$ $-x+8 = -1$ $-x = -9$ $x = 9$ The solution is $\enspace x = 9, \enspace y = -\dfrac{4}{3}$.